December  2012, 2(4): 331-359. doi: 10.3934/mcrf.2012.2.331

Carleman inequalities for the two-dimensional heat equation in singular domains

1. 

Faculté de Mathématiques, laboratoire AMNEDP, U.S.T.H.B, B.P. 32, El-Alia, 16111 ALGER, Algeria, Algeria, Algeria, Algeria

Received  January 2012 Revised  May 2012 Published  October 2012

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we prove a Carleman inequality and we deduce a result of controllability.
Citation: Abdelhakim Belghazi, Ferroudja Smadhi, Nawel Zaidi, Ouahiba Zair. Carleman inequalities for the two-dimensional heat equation in singular domains. Mathematical Control and Related Fields, 2012, 2 (4) : 331-359. doi: 10.3934/mcrf.2012.2.331
References:
[1]

A-H. Belghazi, Null controllability of three-dimensional heat equation for singular domains and stratified media spectral inequality,, in preparation., (). 

[2]

A. H. Belghazi, F. Smadhi, N. Zaidi and O. Zair, Carleman inequalities for the heat equation in singular domains, C. R. Acad. Sci. Paris, 348 (2010), 277-282. doi: 10.1016/j.crma.2010.02.007.

[3]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, On the controllability of linear parabolic equations with an arbitrary control location for stratified media, C. R. Acad. Sci. Paris, 344 (2007), 357-362.

[4]

R. Bey, J. P Lohéac and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation, J. Math. Pures Appl., 78 (1999), 1043-1067.

[5]

L. Bourgeois, A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners, C. R. Acad. Sci. Paris, 345 (2007).

[6]

H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, 1983.

[7]

N. Burq, Contrôle de l'équation des ondes dans des ouverts contenants des coins, Bull. Soc. math. Franc, 126 (1998), 601-637.

[8]

T. Carleman, Sur un problème d'unicit'e pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 1-9.

[9]

A. Doubova, A. Osses and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var., 8 (2002), 621-661.

[10]

H-O. Fattorini and D-L.Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.

[11]

H-O. Fattorini and D-L.Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (): 45. 

[12]

E. Fernandez-Cara and S. Guerrero, Global Carleman Inequalities For Parabolic Systems And application To Controllability, SIAM J. Control Optim, 45 (2006), 1395-1446. doi: 10.1137/S0363012904439696.

[13]

A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[14]

P. Grisvard, Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités, Journal Maths. Pures et Appl., 68 (1989), 215-259.

[15]

P. Grisvard, "Singularities in Boundary Value Problems," Springer-Verlag, 1992.

[16]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Pitman, Boston, London, Melbourne, 1985.

[17]

A. Heibig and M. A. Moussaoui, Exact controllability of the wave equation for domains with slits and for mixed boundary conditions, Discrete and continuous dynamical systems, 2 (1996), 367-386.

[18]

L. Hörmander, "Linear Partial Differential Operators," Springer-verlag, 1963.

[19]

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations, 233 (2007), 417-447. doi: 10.1016/j.jde.2006.10.005.

[20]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential. Equations, 20 (1995), 335-356.

[21]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, to appear in ESAIM Control Optim. Calc. Var., (). 

[22]

M. A. Moussaoui and B. K. Saadallah, Régularité des coefficients de propagation de singularités de l'équation de la chaleur dans un domaine polygonal plan, C. R. Acad. Sci. Paris, 293 (1981), 297-300.

[23]

S. Nicaise, Exact controllability of a pluridimensional coupled problem, Rev. Math. Univ. Complut. Madrid, 5 (1992), 91-135.

[24]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.

[25]

J. Zabczyk, "Mathematical Control Theory: An Introduction," Birkhäuser, 1995.

show all references

References:
[1]

A-H. Belghazi, Null controllability of three-dimensional heat equation for singular domains and stratified media spectral inequality,, in preparation., (). 

[2]

A. H. Belghazi, F. Smadhi, N. Zaidi and O. Zair, Carleman inequalities for the heat equation in singular domains, C. R. Acad. Sci. Paris, 348 (2010), 277-282. doi: 10.1016/j.crma.2010.02.007.

[3]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, On the controllability of linear parabolic equations with an arbitrary control location for stratified media, C. R. Acad. Sci. Paris, 344 (2007), 357-362.

[4]

R. Bey, J. P Lohéac and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation, J. Math. Pures Appl., 78 (1999), 1043-1067.

[5]

L. Bourgeois, A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners, C. R. Acad. Sci. Paris, 345 (2007).

[6]

H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, 1983.

[7]

N. Burq, Contrôle de l'équation des ondes dans des ouverts contenants des coins, Bull. Soc. math. Franc, 126 (1998), 601-637.

[8]

T. Carleman, Sur un problème d'unicit'e pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 1-9.

[9]

A. Doubova, A. Osses and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var., 8 (2002), 621-661.

[10]

H-O. Fattorini and D-L.Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.

[11]

H-O. Fattorini and D-L.Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (): 45. 

[12]

E. Fernandez-Cara and S. Guerrero, Global Carleman Inequalities For Parabolic Systems And application To Controllability, SIAM J. Control Optim, 45 (2006), 1395-1446. doi: 10.1137/S0363012904439696.

[13]

A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[14]

P. Grisvard, Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités, Journal Maths. Pures et Appl., 68 (1989), 215-259.

[15]

P. Grisvard, "Singularities in Boundary Value Problems," Springer-Verlag, 1992.

[16]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Pitman, Boston, London, Melbourne, 1985.

[17]

A. Heibig and M. A. Moussaoui, Exact controllability of the wave equation for domains with slits and for mixed boundary conditions, Discrete and continuous dynamical systems, 2 (1996), 367-386.

[18]

L. Hörmander, "Linear Partial Differential Operators," Springer-verlag, 1963.

[19]

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations, 233 (2007), 417-447. doi: 10.1016/j.jde.2006.10.005.

[20]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential. Equations, 20 (1995), 335-356.

[21]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, to appear in ESAIM Control Optim. Calc. Var., (). 

[22]

M. A. Moussaoui and B. K. Saadallah, Régularité des coefficients de propagation de singularités de l'équation de la chaleur dans un domaine polygonal plan, C. R. Acad. Sci. Paris, 293 (1981), 297-300.

[23]

S. Nicaise, Exact controllability of a pluridimensional coupled problem, Rev. Math. Univ. Complut. Madrid, 5 (1992), 91-135.

[24]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.

[25]

J. Zabczyk, "Mathematical Control Theory: An Introduction," Birkhäuser, 1995.

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